International Journal of Advanced Mathematical Sciences, 1 (2) (2013) 78-86 c Science Publishing Corporation www.sciencepubco.com/index.php/IJAMS On the application of Fubini’s theorem in the integration of functions of two variables in a measure space B.V. Akinremi*, B.N. Akintewe, K.S. Famuagun, U.S. Idiong Adeyemi College of Education, Ondo, Nigeria *Corresponding author E-mail: akinremibv@gmail.com Abstract We consider the integration of functions of two variables in a measure space. Some definitions, theorems and proves relating to measurable functions and measure space were considered by using Fubini’s theorem. Application on the improvement of the Jensen’s inequality with respect to the probability measure space is treated. Keywords : Fubini’s Theorem, Hermite-Hadamard inequality, Jensen’s inequality, Measure, Product measure. 1 Introduction In Mathematical analysis, integration of functions is of great importance. In the 1850’s Bernhard Riemann adopted a new and different viewpoint of the calculus of integration by I. Newton and G. Leibniz. He separated the concept of integration from its companion, differentiation, and examined the motivating summation and limit process of finding areas by itself. He broadened the scope by considering all functions on an interval for which this process of integration could be defined: the class of ’integrable’ functions. The viewpoint of Riemann led others to invent other integration theories, the most significant being Lebesgue’s thoery of integration. Measure theory was developed in successive stages during the late 19th- early 20th century by Emile Borel, Henri Lebesgue, Johann Radon among others. In measure integration theory, specifying a measure allows one to define integrals on spaces more general and it gives more theorem than its predecessor, the Riemann integral. Also, in classical analysis the problem of reducing double(or multiple) integrals to iterated integrals plays an important role. In integration theory, the key result along this line is the Fubini’s theorem. Fubini’s theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to com- pute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals. In 1992 Dragomir and Ionescu [6] studied some aspects of convex functions and some interesting inequalities were obtained. Mitrinovic, Pecaric and Fink in [12] discussed classical and new inequalities in analysis like the Holder’s, Minkowski’s, Jensen’s, Bernoullli’s and Steffensen’s inequalities in which they proved these inequalities with some generalizations. Dragomir and Ionescu [8] later in 1994 proved useful inequality which counterparts the Jensen’s inequality for continuous functions. Rooin[14] proved some application on discrete function. 2 Preliminary notes We briefly give some basic definitions of the concepts of measure, integration and product space which serves as background to this work. Definiton 2.1.0[11] A system of set is a set whose elements are sets. A system of set I is called a semiring if